Inertia-gravity-wave diffusion by geostrophic turbulence: the impact of flow time dependence

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad et al. (J. Fluid Mech., vol. 869, 2019, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.

Automated summary

The dispersion of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to a diffusion process in wavevector space. Previous studies have obtained the diffusivity tensor assuming a time-independent geostrophic flow. This study relaxes this assumption and examines how the weak diffusion of wave action affects the distribution of wave energy due to the slow time dependence of the geostrophic flow. The results show that the wave-energy spectrum is localized within a thin boundary layer around the constant-frequency cone, with its thickness controlled by the acceleration spectrum of the geostrophic flow.